Classifying space for SO(n) explained

In mathematics, the classifying space

\operatorname{BSO}(n) for the special orthogonal group \operatorname{SO}(n) is the base space of the universal \operatorname{SO}(n) principal bundle \operatorname{ESO}(n) → \operatorname{BSO}(n) . This means that \operatorname{SO}(n) principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into \operatorname{BSO}(n) . The isomorphism is given by pullback.

Definition

There is a canonical inclusion of real oriented Grassmannians given by

\widetilde\operatorname{Gr} ^{k +1} _{n( R} ), V\mapsto V x \{ 0 \} . Its colimit is:^[1] \operatorname{BSO}(n) : = \operatorname{Gr} ^{inf ty) : = \lim} _{k → inf ty} \widetilde\operatorname{Gr} ^{k).} _{n( R}

Since real oriented Grassmannians can be expressed as a homogeneous space by:

\widetilde\operatorname{Gr} ^{k) = \operatorname{SO}(n + k)/(\operatorname{SO}(n) x \operatorname{SO}(k))} _{n( R}

the group structure carries over to

\operatorname{BSO}(n) .

Simplest classifying spaces

• Since

\operatorname{SO}( 1 ) \cong 1 is the trivial group, \operatorname{BSO}( 1 ) \cong\{*\} is the trivial topological space.

• Since

\operatorname{SO}( 2 ) \cong\operatorname{U}( 1 ) , one has \operatorname{BSO}( 2 ) \cong\operatorname{BU}( 1 ) \cong C P ^{infty} .

Classification of principal bundles

X the set of \operatorname{SO}(n) principal bundles on it up to isomorphism is denoted \operatorname{Prin} _{\operatorname{SO} (n)}(X) . If X is a CW complex, then the map:^[2] [X,\operatorname{BSO}(n)] → \operatorname{Prin} _{\operatorname{SO} (n)}(X), [f]\mapsto f ^{*\operatorname{ESO}(n)}

is bijective.

Cohomology ring

The cohomology ring of

\operatorname{BSO}(n) with coefficients in the field Z _{2} of two elements is generated by the Stiefel–Whitney classes:^{[3] [4]} H ^{*(\operatorname{BSO}(n); Z} _{2 ) = Z} _{2[w} _{2,\ldots,w} _{n].} \operatorname{char} =2 .

The cohomology ring of

\operatorname{BSO}(n) with coefficients in the field Q of rational numbers is generated by Pontrjagin classes and Euler class:H ^{*(\operatorname{BSO}( 2 n); Q ) \cong Q [p} _{1 ,\ldots,p} _{n,e]/(p} ^{2 ),} ^{n - e} H ^{*(\operatorname{BSO}( 2 n +1 ); Q ) \cong Q [p} _{1 ,\ldots,p} _{n].}

The results holds more generally for every ring with characteristic

\operatorname{char} ≠ 2 .

Infinite classifying space

The canonical inclusions

\operatorname{SO}(n)\hookrightarrow\operatorname{SO}(n +1 ) induce canonical inclusions \operatorname{BSO}(n)\hookrightarrow\operatorname{BSO}(n +1 ) on their respective classifying spaces. Their respective colimits are denoted as:\operatorname{SO} : = \lim _{n → infty} \operatorname{SO}(n); \operatorname{BSO} : = \lim _{n → infty} \operatorname{BSO}(n). \operatorname{BSO} is indeed the classifying space of \operatorname{SO} .

See also

• Classifying space for O(n)
• Classifying space for U(n)
• Classifying space for SU(n)

Literature

• Book: Milnor, John. Characteristic classes. Princeton University Press. 1974. 9780691081229. 10.1515/9781400881826. John Milnor. Stasheff. James. James Stasheff.
• Book: Hatcher, Allen. Algebraic topology. Cambridge University Press. Cambridge. 2002. 0-521-79160-X.
• Book: Mitchell, Stephen. Universal principal bundles and classifying spaces. August 2001.

External links

• classifying space on nLab
• BSO(n) on nLab

References

1. Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151
2. Web site: 2024-03-14 . en . universal principal bundle . nLab.
3. Milnor & Stasheff, Theorem 12.4.
4. Hatcher 02, Example 4D.6.